If the radicals have the same index,
a) then, multiply the coefficients
b) multiply the radicand
c.) Simplify
Example:
[tex](5 \sqrt[3]{x} ) ( \sqrt[3]{ x^{2} } )[/tex] ⇒ they have the same index, 3
a) (5) (1) = 5
b) [tex] (\sqrt[3]{x} )( \sqrt[3]{x ^{2} } ) = \sqrt[3]{ x^{3} } [/tex]
c) The answer is [tex]5 \sqrt[3]{x^{3} } = 5x[/tex]
If the radicals are not similar, not having same index:
Example:
[tex]( \sqrt[3]{ x^{2} }) ( \sqrt{x ^{3} } )[/tex]
The first radical has index of 3; the second has 2.
a) Convert the radical to rational exponents:
[tex] \sqrt[3]{ x^{2} } = x ^{ \frac{2}{3} } [/tex]
[tex] \sqrt{x ^{3} } = x \frac{3}{2} [/tex]
b) Convert fractional exponent to similar fractions; Find their LCD⇒6
[tex]x \frac{2}{3}= x \frac{4}{6} [/tex]
[tex]x \frac{3}{2}=x \frac{9}{6} [/tex]
c) Convert to radicals then multiply. The index for both radicals is 6.
[tex]( \sqrt[6]{x^{4} } )( \sqrt[6]{x^{9} } ) = \sqrt[6]{x^{13} } [/tex]
d) Simplify:
[tex] \sqrt[6]{x^{13} } = x^{2} \sqrt[6]{x} [/tex]
